Ihor Kendiukhov

The post covers a lot of ground, but the section relevant to our discussion is section 5, titled The Independence Axiom Isn't So Bad.

Akademian's defense of independence rests on what he calls the contextual strength, CS, interpretation of VNM utility.

The idea is that VNM preference should be understood as strong preference within a given context of outcomes.

When the VNM formalism says you are indifferent between two options, S equals D in the parent giving a car to children example, this does not mean you have no preference at all.

It means you have no preference strong enough that you would sacrifice probabilistic weight on outcomes that matter in the current context in order to indulge it.

Under this interpretation, the independence axiom's requirement that S equals D implies S equals F equals D, where F is the coin flip mixture, just means you wouldn't sacrifice anything contextually important to get the fair coin flip over either deterministic option.

You can still prefer the coin flip in some weaker sense.

You just can't prefer it strongly enough to trade off against the things that actually matter.

I want to acknowledge that this is a well-crafted defense, and Akademian is admirably honest about most of its limitations.

But the CS defense has a critical limitation that Akademian does not address.

It works only for small, contextually negligible independence violations.

The parent and car example involves a marginal preference for fairness that is, as Akademian argues, plausibly too weak to warrant probabilistic sacrifice in a context that includes weighty outcomes.

Fine.

But the independence violations that arise in the settings this article is concerned with are not marginal at all.

Consider again the gamble example from Section 3.

You are choosing between gambles A and B, and the common component C is either a large safety net, 10 million euros, or a trivial amount, 5 euros.

Your preference between A and B flips depending on what C is.

With a large safety net, you take the risky option.

Without it, you take the safe one.

This is not a whisper of a preference that disappears when larger considerations are in play, but a robust, large-magnitude shift in risk strategy driven by the structural properties of your total exposure.